Effect of proppant injection condition on fracture network structure and conductivity

Abstract

The hydraulic fracturing of shale reservoirs is an important topic, and adjusting the fracturing process can enhance fracturing efficiency. To analyze the effect of injection conditions on the fracture network structure and conductivity, numerical simulations and theoretical calculations are carried out based on the construction data, and the variation patterns of the fracture network parameters are determined. During the theory study, a fracture network conductivity model was obtained by combining the fracture network expansion model and the fracture diversion model. Then a numerical model was set up according to the analysis results of monitoring data. The numerical simulation results showed the fracture network structure and proppant distribution under different conditions. Throughout the simulation process, the structural parameters and conductivity of the fracture network can be determined by controlling the injection amount and adjusting the injection method, speed, and sand ratio. Results indicated that the proppant pulse frequency during the injection process was a major factor affecting the structure and conductivity of the fracture network. At the same time, a comparison of the distribution structure of the proppant revealed that although the intermittent injection of the proppant improved the flow channel, the efficiency of fracture utilization can still be improved.

INTRODUCTION

Injection condition not only affects proppant distribution in fractures but also changes the structure and conductivity of fracture network. Intermittent injection can cause pressure fluctuation inside fractures, which enhances fluid erosion on the fracture walls, thereby influencing the expansion of the fracture network. The expansion of the fracture network changes the permeability of the reservoir, enhancing the conductivity of fractures. Herein, fracture network structure and conductivity are analyzed via numerical simulation and theoretical calculation to understand the impact of intermittent injection on the effectiveness of hydraulic fracturing treatments.

To study the effect of injection methods on the network structure and conductivity of fractures, researchers have analyzed the factors involved in the injection methods, such as pump time ratios, injection volumes, and proppant concentrations, by combining theoretical and experimental studies.^1,2 The analysis results show that the injection method has a significant effect on the effectiveness of fracture network modification. Gillard et al.³ reported that intermittent injection of the proppant into fracture during the fracturing process can considerably improve the conductivity of fractures. A visualization experiment is designed by Medvedev et al.⁴ based on on-site construction parameters to study intermittent injection. Their results showed that the settling trajectory of the proppant is disturbed in fracture, resulting in the nonuniform distribution of proppant dams within fractures. On the basis of similarity experimental, Yang and Wen⁵ established a calculation model for relevant parameters, such as bottom hole flowing pressure, the viscosity and density of the mixture in the reservoir, and the mass flow rate.

Rock damage performance analysis shows that pulse fluid injection can enhance fracture expansion performance. He et al.⁶ found that the pulse fracturing of shale reservoirs at a higher fluid pressure gradient requires a shorter time, making it easier to form complex fractures. Compared with stable injection methods, intermittent injection can markedly reduce the maximum wellhead pressure from Zhu et al.⁷ The expansion structure of the fracture network is also influenced by the injection method. Di et al.⁸ established a multifracture intersecting fracture extension seepage model by combining the mechanism of water-induced fracture extension and principles of fracture mechanics energy conservation to obtain the length of fracture extension. On the basis of the global embedded three-dimensional (3D) cohesive zone model, Hu et al.⁹ developed a geometric simulation model of the fracture network in a fractured, porous, elastic reservoir to study the effect of the injection method on the structure of the fracture network. Zeng et al.¹⁰ used the displacement discontinuity method and Picard iteration method to establish a fracture expansion model by considering the interaction between hydraulic and natural fractures.

Intermittent injection changes the distribution pattern of the proppant in fracture, enhancing the conductivity of the fracture. Low-density proppants were more likely to form an open-flow channel network than high-density proppant; Liang¹¹ observed that fiber mixing was the most notable factor influencing channel conductivity. This is because the fibers contribute to the formation of clusters within the fracture, Zhao et al.¹² studied the effect of different factors (such as proppant diameter, fluid viscosity, fiber concentration, and flow rate) on the structure of proppant clusters. On the basis of the API test, Cao et al.¹³ analyzed the effects of the proppant cluster structure, size, and arrangement on fracture conductivity and observed changes in the cluster support structure due to closure stress. Lu et al.¹⁴ simulated the deformation patterns of proppant clusters under different closure stress conditions and calculated fracture conductivity based on the cluster deformation pattern. Yang et al.¹⁵ and Jin et al.¹⁶ conducted experimental tests to understand the effect of factors such as the placement method, proppant concentration, mixed fiber concentration, proppant density, and proppant size on the conductivity of fracture.

By studying the relation between injection methods and structures of fracture networks, the influence of the structure on the conductivity of fracture networks can be analyzed. Ji et al.¹⁷ calculate the conductivity of fracture networks based on the principles of hydraulic similarity, comprehensively considering the effects of network fracture and fluid types, proppant composition ratios, and placement methods. Zhao et al.¹⁸ established a multilevel fracture conductivity optimization model and analyzed the conductivity of self-supporting and sandstone-supported fractures. Ye et al.¹⁹ developed a systematic method to understand the relation between the permeability and connectivity of two-dimensional fracture networks. A nonlinear flow reduced the effective conductivity of the fracture network, Zhu²⁰ showed that the correlation between aperture and fracture lengths increased the effective conductivity. Zhao et al.²¹ analyzed the impact of the main fracture geometry and distribution parameters of secondary fractures on production. These research results explain the diversion ability of the fracture network from different methods.

In addition, a change in the proppant distribution changes the propped fracture height, which affects fracture conductivity.²² The effect of natural fractures on the productivity of the process was studied via numerical simulations,²³ in which the calculation accuracy was improved by refining the structure in the fracture network.²⁴ Irregular-shaped fracture expansion is also the main factor affecting fracture conductivity,²⁵ and fractures in the network interfere with each other during the fracture expansion process.^26,27

It can be found that the previous research still has shortcomings. On the one hand, the mechanism analysis of complex fracture network structure is relatively simple. On the other hand, the calculation model of the conductivity of complex fracture networks is relatively lacking. Through numerical simulation and theoretical research, this paper supplements the content of these two aspects.

THEORETICAL RESEARCH OF NUMERICAL SIMULATIONS

Calculating the parameters of the fracture network structure

The structure of the fracture network has two levels of meaning. First, it refers to the changes in the distribution structure of the fracture network. Second, it involves the changes in the geometric structure of individual fractures within the network. When studying the changes in the distribution structure parameters of the fracture network, the variations in the length, width, and network density of the fractures are analyzed. When investigating the geometric structure parameters of individual fractures within the fracture network, the analysis is performed using the extension of the main fracture, the structure of the fracture network, and the distance between fractures.

According to the relevant literature^24,28 on traditional fracture expansion models, the main fracture follows the law of quasi-3D fracture expansion during the formation of fracture. The following assumptions are made: (1) The reservoir is a continuous elastic body with a large thickness, and fractures only propagate within the reservoir. (2) Fracturing fluid is injected into the fracture with a constant displacement, and the flow of the fracturing fluid in the fracture is one-dimensional along the length of the fracture, with laminar flow. (3) The main fracture is transverse in the horizontal section perpendicular to the well, symmetrically distributed with the wellbore as the axis. (4) The placement of the proppant in fracture and the influence of temperature on the properties of the fracturing fluid are not considered.

In Figure 1, a is the half-length of the main fracture (m), b is the maximum extension half-length of the fracture network in the Y-axis (m), and d_x and d_y are the secondary suture intervals perpendicular to the X- and Y-axes (m), respectively.

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Equations for the structure of secondary fractures are obtained using the structure of the main fracture and the relation set between the secondary and main fractures. An equation for determining the number of secondary fractures is derived using the relation between the fracture spacing and the semimajor axis length of an ellipse. 1$$\left\{\begin{array}{c}{N}_{x}=\frac{2a}{{d}_{x}}-1,\\ {N}_{y}=\frac{2b}{{d}_{y}}-1,\end{array}\right.$$where N_x and N_y are the numbers of secondary fractures perpendicular to the X- and Y-axes, respectively, and b = γa. On the basis of the position of the secondary fractures in fracture and the length of the main fracture, the parameters of secondary fractures are calculated. 2$$\left\{\begin{array}{c}{L}_{xs}=2\sum _{i=1}^{{N}_{x}/2}{L}_{xis}=2\sum _{i=1}^{{N}_{x}/2}\left[\frac{1}{\gamma }\sqrt{{a}^{2}-{\left(i-\frac{1}{2}\right)}^{2}{d}_{x}^{2}}\right],\\ {L}_{ys}=2\sum _{j=1}^{{N}_{y}/2}{L}_{yjs}=2\sum _{j=1}^{{N}_{y}/2}\left[\frac{1}{\gamma }\sqrt{{b}^{2}-{\left(j-\frac{1}{2}\right)}^{2}{d}_{y}^{2}}\right],\end{array}\right.$$where L_xs and L_ys are the total lengths of secondary fractures perpendicular to the X- and Y-axes (m), respectively, L_xis is the fracture length of the ith secondary fracture perpendicular to the X-axis (m), and L_yjs is the fracture length of the jth secondary fracture perpendicular to the Y-axis (m). Combined with the calculated results of Equation (1) and the parameters of the fracture network, the secondary fracture length can be obtained through Equation (2). This equation aims to divide the fracture into several segments along its length and then solve for the fracture width using the method of plane strain problems. Here, it is assumed that the main fracture only extends within the reservoir, and by simplifying the theoretical model of England and Green, an expression for calculating the width of the main fracture is obtained. 3$${w}_{D}=\frac{2(1-{v}^{2})}{E}({p}_{f}-{\sigma }_{\min }){h}_{D},$$where v is the dimensionless Poisson's ratio of the reservoir, E is the elastic modulus of the reservoir (MPa), p_f is the fluid pressure in the main fracture (MPa), and σ_min is the minimum horizontal principal stress in the reservoir (MPa). The cross-section of the fracture network is elliptical, and the distance of the fracture is uniformly distributed in the directions along the X- and Y-axes, following the elliptical formula, an average fracture width ($${\bar{w}}_{D}$$) (calculated using Equation 3, which is substituted into Equation 4) is used to characterize the width of the main fracture for the convenience of subsequent model calculations. 4$${\bar{w}}_{D}=4\pi /{w}_{D}.$$

The secondary fracture width is obtained by substituting the w_D value obtained in Equation (3) into Equation (5): 5$$\left\{\begin{array}{c}{w}_{xs}={\lambda }_{x}{w}_{D},\\ {w}_{ys}={\lambda }_{y}{w}_{D},\end{array}\right.$$where w_xs and w_ys are the widths of secondary fractures perpendicular to the X- and Y-axes (mm) and λ_x and λ_y are the ratios of the widths of secondary fractures perpendicular to the X- and Y-axes, to the width of the main fracture, respectively.

According to the fracture expansion criterion in fracture mechanics, the governing equation for calculating the fracture height is established. According to the theory of linear elastic fracture mechanics, the stress intensity factor K_I1 at the fracture tip is calculated as follows: 6$${K}_{I1}=\frac{1}{\sqrt{\pi l}}{\int }_{-l}^{l}({p}_{f}-{\sigma }_{\min })\sqrt{\frac{l+z}{l-z}}\unicode{x0200A}dz,$$where K_I1 is the stress intensity factor (MPa m^1/2) and l is the half-fracture height (m). According to fracture mechanics, when the stress intensity factor at the fracture tip reaches the fracture toughness of the fracture, the fracture starts to propagate. 7$${K}_{IC1}={K}_{I1},$$where K_IC1 is the fracture toughness of the fracture (MPa m^1/2). Combining Equations (6) and (7), the fracture height is obtained as follows: 8$${h}_{D}=2l=\frac{2{K}_{IC1}^{2}}{{({p}_{f}-{\sigma }_{\min })}^{2}\pi }.$$

It is assumed that the hydraulic fracturing transformation volume of the fracture network is an ellipsoid, and the distribution of the secondary fracture height satisfies an elliptical relation with the height of the main fracture. The secondary fracture height is calculated using Equation (9): 9$$\left\{\begin{array}{c}{h}_{sxi}=\frac{{h}_{D}}{a}\sqrt{{a}^{2}-{\left(i-\frac{1}{2}\right)}^{2}{d}_{x}^{2}},\\ {h}_{syj}=\frac{{h}_{D}}{b}\sqrt{{b}^{2}-{\left(j-\frac{1}{2}\right)}^{2}{d}_{y}^{2}},\end{array}\right.$$where h_sxi is the height of the ith secondary fracture perpendicular to the X-axis (m) and h_syj is the height of the jth secondary fracture perpendicular to the Y-axis (m). The average height of the fractures in the fracture network is obtained using an ellipsoidal volume with the same a and b values as an equivalent to the ellipsoidal volume, the average fracture height can be obtained using Equation (10). 10$$\bar{h}=\frac{2}{3}{h}_{D}.$$

Ignoring the influence of fracture closure on the structure of the fracture network and focusing primarily on the effect of variations in fracture width, a model for calculating the conductivity of the fracture network is established. The influence of injection methods on the conductivity is analyzed. The distribution of the fracture network can be obtained by combining microseismic monitoring data and numerical simulation results. Therefore, the parameters for calculating the conductivity of the fracture network need to be adjusted based on the actual situation.

Conductivity model of fracture network

Assuming that the closure pressure of fractures remains constant and neglecting factors such as proppant crushing and proppant embedment, a numerical model for the conductivity of a single fracture is established under certain assumptions. On the basis of the capillary tube model and the Carman–Kozeny equation, a formula for calculating the permeability of a single fracture is derived.^17,18,29 11$${K}_{w}=\frac{\left[LHW-4N\pi {r}_{p}^{3}/3\right]{r}^{2}\times {10}^{-15}}{8LHW{\tau }^{2}},$$where K_w is the permeability of a single fracture (mD), L is the length of the fracture (m), H is the height of the fracture (m), W is the width of the fracture (m), N is the total number of proppants placed in the unit fracture area, r_p is the radius of the proppant (m), r is the pore radius of the propped fracture (m), and τ is tortuosity, with a value of 1.1547.

The calculation of the conductivity of a fracture network is quite complex and requires a comprehensive analysis that considers the diversion through branched fractures and single-fracture seepage. Without considering the influence of natural fractures and matrix permeability, the conductivity of the fracture network is calculated based on the principles of hydraulic similarity.

Figure 2 shows that the permeability resistance of the fracture network needs to be calculated in parallel. The permeability of a single fracture can be obtained using Equation (11), and the flow resistance of a single fracture can be calculated by substituting K_w in Equation (12): 12$${R}_{k}=\frac{\mu L}{{K}_{w}H}\unicode{x02007}\unicode{x02007}(k=1,2,3,{\rm{\ldots }},3j+1).$$

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Where R_k is the permeability resistance of each fracture (mPa s/mD). After calculating R_k using Equation (12), the appropriate Equation (13) can be selected according to the fracture condition parameters. According to the parallel principle, the permeability resistance of the fracture network is expressed as follows: 13$$\left\{\begin{array}{c}{R}_{t1}=\frac{1}{\displaystyle \frac{1}{{R}_{1}}+\displaystyle \frac{1}{{R}_{2}}+\displaystyle \frac{1}{{R}_{3}}}\unicode{x02007}\unicode{x02007}(j=1),\\ {R}_{tj}=\frac{1}{\displaystyle \frac{1}{{R}_{t(j-1)}+{R}_{3j-2}}+\displaystyle \frac{1}{{R}_{3j-1}}+\displaystyle \frac{1}{{R}_{3j}}}\unicode{x02007}\unicode{x02007}(j\ge 2),\\ {R}_{FN}={R}_{tj}+{R}_{3j-2}.\end{array}\right.$$

Equation (13) can be used to calculate the flow resistance of different fracture network structures. Using the calculated permeability from the permeability resistance of the fracture network, the conductivity of the fracture network is derived. 14$${K}_{FN}=\frac{\mu L}{{R}_{FN}{W}_{FN}H},$$ 15$${F}_{FN}={K}_{FN}{W}_{FN}=\frac{\mu L}{{R}_{FN}H},$$where K_FN is the fracture network permeability (mD), W_FN is the equivalent fracture width of the fracture network (m), and F_FN is the fracture network conductivity (mD m). The conductivity of the fracture network can be obtained by combining Equations (14) and (15).

When analyzing the influence of the fracture structure parameters, the irregularities on the fracture surfaces are ignored. On the basis of the calculation of the conductivity of the fracture network, the impact of injection methods on the improvement effect is analyzed.

APPLICATION SITUATION

The application data could be organized and analyzed to provide simulation parameter settings and validate simulation results. To verify the comparative results, the data in this study were collected from different fracturing well groups on the same platform, ensuring identical development conditions.

Construction design of fracturing

An oilfield block having seven horizontal wells on the 3# Platform was chosen, targeting the Shi Niulan Formation and the Lower Ordovician Wufeng Formation shale gas reservoir section. To analyze the impact of injection conditions on reservoir transformation results, three fracturing wells with distinctive injection methods were selected, fracturing parameters shown in Table 1. For well 3HF, the proppant combination was 40/70 mesh + 70/140 mesh low-density ceramic beads and the fracturing fluid combination includes high- and low-viscosity slickwater and a gel. For wells 3-3HF and 3-6HF, the proppant combination is 40/70 mesh + 70/140 mesh + 140/260 mesh quartz sand, and the fracturing fluid combination includes acid, low-viscosity slickwater, and a gel. The fracturing process adopted a segmented multicluster fracturing technique; therefore, sections with the same perforation parameters were selected for comparative analysis.

Table 1 Fracturing parameters.

Number	Number of cluster/perforations	Proppant volume (m3)	Fracturing fluid volume (m3)
40/70	70/140	140/260	Total	Low	High	Glue	Acid	Total
3HF4	4/48	46.38	25.21	–	71.59	507	896.15	195	–	1598.15
3-3HF1	4/48	17	52.3	45.7	115	1376.2	–	32.8	20	1429
3-6HF1	4/48	14	67	19.2	100.2	1650.2	–	97	20	1767.2

The construction of the three fracturing stages, that is, 3HF4, 3-3HF3, and 3-6HF3, proceeded smoothly, and no faults were encountered during fracturing. By combining the configurations, functions, and purposes of the construction parameters, the characteristics of different injection methods were analyzed. Continuous injection of proppant in fracture requires the use of an acid to clean fractures, and the acid can simultaneously enlarge the size of fractures or pores in fracture. The extensive use of a low-viscosity fracturing fluid can facilitate the rapid expansion of pressure within the fractures; however, the efficiency of pressure transmission is relatively low. During the continuous injection of the proppant into the fracture, a small-diameter proppant is initially used to help expand fractures; then, a large-diameter proppant is injected to maintain support. The effectiveness of intermittent injection is considerably influenced by the suspension performance of the fracturing fluid, leading to greater use of the high-viscosity fracturing fluid and gel. Meanwhile, the gel and high-viscosity fracturing fluid can achieve pressure balance after fracturing, maintaining the stability of the fractures. Because accurate evaluation of the impact of injection methods on the structure of the fracture network through differences in construction parameters, we combine the fracturing data with simulation results to analyze the structure and conductivity of the fracture network under different injection conditions.

Microseismic monitoring technology can be used to determine the positions, orientations, and connectivity of fractures within the fracture network, which helps establish a more accurate discrete fracture network (DFN) model. The analysis of fracture monitoring data in Table 2 allows one to obtain the structure of the fracture network after fracturing.

Table 2 Microseismic monitoring results.

Number	Length (m)	Width (m)	Height (m)	Orientation angle (°)
3HF4	385	83	40	127
3-3HF1	185.143	61.7143	41.1429	112
3-6HF1	284.393	83.237	41.6185	113

Pump injection parameters, fracture monitoring data, and hydraulic fracturing construction curves were combined to analyze the injection methods and fracturing characteristics of the three fracturing stages. In the case of Figure 3A, the pulsation frequency of the sand-carrying fluid is relatively high. It not only cleans the near-wellbore zone but also helps expand the reservoir transformation area. Throughout the injection process, the overall viscosity of the fracturing fluid in 3HF4 is relatively high, with a relatively low proppant concentration. For the case of Figure 3B,C, acid is employed to clean the near-wellbore zone, and the injection process primarily uses the low-viscosity fracturing fluid. Regarding the injection methods, the early stage of 3-6HF3 involves intermittent injection, followed by a high-frequency sand-carrying fluid stage, resulting in a larger stimulated reservoir volume (SRV). The 3-3HF3 can be approximated as a continuous injection, featuring a higher proppant concentration but a smaller SRV.

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Construction parameter analysis

On the basis of the operation parameters, the initiation and closure states of fractures can be analyzed. The changing trends can be analyzed to understand the variation in stress during the fracturing process. The simulator, using fundamental reservoir data and playback of pump injection parameters, calculates the variation in stress in fractures during the initiation and closure processes. The results are shown in Table 3. Ultimately, the accuracy of the simulation results is improved by correcting the input data of the fracture model.

Table 3 Fracturing test analysis results.

Number	Fracture extension pressure (MPa)	Extended pressure gradient (MPa/m)	Fracture closure pressure (MPa)	Pore pressure gradient (MPa/m)
3HF4	58.3	0.024	38.34	0.024
3-3HF1	96.7	0.039	55.32	0.027
3-6HF1	108.4	0.039	52.06	0.026

On the basis of stress analysis, the G-function can be used to evaluate the fracture network. In shale reservoir-fracturing projects, due to the tight nature of the rock, filtration can be regarded as the degree of microfracture richness. Therefore, the complexity of the fractures can be analyzed using the G-function. The main characteristics of fractures formed by hydraulic fracturing include high shut-in pump pressure, high friction, and high filtration. If multiple artificial fractures simultaneously extend, the fracture extension pressure will continue to increase after rock fracturing, appearing as a pronounced concave segment on the G-function curve. Owing to the overall small differences in the filtration coefficients of the reservoir matrix, the G-function response curve of shale formations with undeveloped microfractures is a straight line. Generally, a high-slope G-function response curve corresponds to higher matrix permeability. When microfractures develop within the formation, the G-function response curve exhibits a noticeable upward convexity.^30,31

The degree of fracture development can be briefly assessed using the data of fracture monitoring and liquid efficiency. The effectiveness of hydraulic fracturing can be roughly evaluated using the monitoring data of production parameters in Figure 4. By comprehensively analyzing these data, the final distribution of the fracture network and natural fractures in fracture can be determined. Data analysis results indicate that the development of natural fractures is relatively poor in 3HF4 but is better in 3-3HF1 and 3-6HF1. However, G-function analysis reveals that the fracturing effect of 3HF4 is better than those of 3-3HF1 and 3-6HF1, demonstrating the advantages of intermittent injection.

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Model validation

Microseismic monitoring technology can be used to predict the development trend and distribution area of fracturing fractures and then track and evaluate the fracturing construction effect. The microseismic monitoring effect of 3HF is shown in Figure 5. A total of 20 stages of monitoring have been completed, and 916 microseismic events have been monitored cumulatively. The grid size of the cell in the background of the figure is 100 m × 100 m.

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This project adopts the large-stage multicluster fracturing. Different colors in Figure 5 represent different stage positions, and the fracture network parameters can be known through the processing of monitoring data. A single-stage extension of a hydraulic fracturing average event is about a length of 419 m, a width of 62 m, a height of 40 m high, and the bearing range of the event extension is between 114° and 134°.

The simulation results of stage 4 are shown in Figure 6, which are divided into four clusters. To verify the accuracy of the simulation results of the fracture network structure, the structure parameters of the fracture network are compared with the microseismic monitoring data.

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The verification results are shown in Table 4, it can be seen that the average error of the geometric parameters is 0.328%, which proves that the simulation results are basically consistent with the monitoring situation.

Table 4 Comparison of structure parameters.

3HF4	Length (m)	Width (m)	Height (m)	Azimuth (°)
Monitoring result	385	83	40	127
Simulation Result	386.9	83.21	40.095	127

SIMULATION RESULT ANALYSIS

Simulation parameter setting

Production parameter setting

Herein, hydraulic fracturing construction parameters and monitoring parameters were used to establish a numerical model. The model was used to predict the transport rule of the proppant within fractures. By adjusting injection parameters, the accuracy of the simulation results was enhanced, and the pressure error during the stable injection stage was controlled <5%. Sensitivity parameter ranges were set from the perspective of on-site process requirements, with the maximum flow rate change ranging from 14 to 22 m³/min and total sand ratio change ranging from 3% to 6%.

The pump parameters maintained the same range in Table 5, and the physical properties of the fracturing fluid and proppant referred to commonly used field configurations. Common types of proppants include quartz sand and ceramic beads, and common types of fracturing fluids include slickwater, linear gel, and gelled fluids. Because of the frequent mixing and matching of proppants and fracturing fluids during the fracturing process, the influences of specific parameters could not be accurately analyzed through sensitivity parameter analysis. Therefore, this study primarily focuses on the analysis of pump parameters and injection methods.

Parameter setting of fracture network

Table 5 Production parameter setting.

Injection flow rate (m3/min)	Injection sand ratio (m3/m3)
12, 14, 16, 18, 20, 22, 24	3.36, 3.92, 4.48, 5.04, 5.6, 6.16, 6.72

Herein, a DFN model was adopted and combined with fracture monitoring data for analysis. The DFN exhibits characteristics such as anisotropy, multiscale features, and discontinuity, providing a more realistic reflection of the fracture structure. Therefore, it is possible to evaluate the impact of different injection methods on fracturing effects from the perspective of fracture network characteristics. Anisotropy is manifested as fracture density, fracture orientation, fracture permeability, and fracture size. Multiscale features involve multiscale effects, scale fractal characteristics, scale-dependent permeability, and multiscale modeling. Discontinuity characterization mainly includes spatial heterogeneity, discontinuity, structure differences, and fracture permeability.

The fracture network parameters of numerical simulation are shown in Table 6. Disregarding factors such as wall roughness and embedment, which may affect the proppant, the frictional resistance of the fracturing fluid was related to the Reynolds number. Because of the large area of the fracture network, the interfracture interference phenomenon occurs, which affects reservoir stress and filtration. The matrix permeability of shale reservoirs is much lower than that of natural fractures; therefore, fluid filtration can be equivalent to the degree of natural fractures. Combining the results of small-pressure analysis with fracturing data, the relation between the distribution of natural fractures and the efficiency of the fracturing fluid was analyzed. The simulation results show that in locations where natural fractures are developed, the efficiency of the fracturing fluid transportation is low. In locations where natural fractures are not developed, the efficiency of the fracturing fluid transportation is high. The spacing between fractures in the fracture network was set according to microseismic detection data.

Table 6 Fracture network parameters setting.

Distance of perforation cluster (m)	Permeability of reservoir (mD)	Perforation number/diameter (mm)
12	0.218	12/9.5
12	0.218	12/9.5
10	0.218	12/9.5

Influencing factors of fracturing parameter

Effect of injection method on fracture network structure

A fracture network model was established using fracture monitoring data, and the changing patterns of the fracture network were analyzed using simulation results. Numerical simulation results visually demonstrate the characteristics of the fracture network. Figure 7 shows the fracture network volume and SRV for the injection method of three stages. Figure 7 shows that the volumes of the fracture network in 3HF4, 3-3HF3, and 3-6HF3 are similar, but there is a clear difference in SRVs. This is attributed to variations in the distribution patterns of the fracture network.

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The fracture network structure in reservoir is shown in Figure 8, with the perforation location in the middle. The simulation results of the fracture network structure are in general agreement with fracture monitoring data, with an average error of <5%. A study of the ratio of the widths of main and secondary fractures indicates that in shale reservoirs, the empirical coefficient range is between 0.1 and 0.5, consistent with simulation results. The distribution of secondary fractures in reservoir was simulated by monitoring the production data from an actual oilfield.

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Figure 8 shows that different injection methods considerably impact the distribution structure of the fracture network. The main fractures of 3HF4 and 3-6HF1 are longer, and the development of secondary fractures is better than those of 3-3HF1. The fracture network of 3-3HF1 is poorly developed, but the individual fracture width is larger than those of 3HF4 and 3-6HF1. A comparison presented in Figure 8A–C shows that intermittent injection forms more branching fractures and larger distribution areas, creating a multilevel fracture network in reservoir. Conversely, continuous injection typically produces wider and relatively continuous fractures. Differences in reservoir remodeling volumes are created by the influence of different injection methods on the distribution structure of fractures in fracture.

The distribution structure of fractures shows that the density of horizontal fractures is significantly greater than that of vertical fractures. Taking the perforation zone as the center, the width of fractures gradually decreases from the inside of the zone to the outside. On the basis of the overall reservoir remodeling effect, it can be observed that DFNs exhibit self-similarity, that is, similar fracture distributions and structures can be seen at any scale. Therefore, to more accurately describe and simulate the behavior of DFNs, multiscale modeling methods are typically employed.

Effect of flow rate on fracture geometry

When analyzing the sensitivity parameters in this paper, the original parameters are taken as the benchmark for equal scaling. To analyze the impact of intermittent injection on fractures, the pump parameters were adjusted using fracture geometry parameters of the main fracture as representation, and the simulation results were analyzed. The maximum flow rate in the injection process is used as the identification point. The expansion pattern of the fracture structure exhibits similarity. By comparing the geometric parameters of the main fractures and evaluating the influence of injection displacement on the structure of the fracture network, Figure 9 shows the change in the geometric parameters of the main fractures in network.

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As the flow rate of the fracturing fluid increases, the length and the height of fractures increase, and the width of fractures decreases. When the flow rate increases from 12 to 22 m³/min, the length increases by 14.6%, and the width of fractures decreases by 25.2%. The change in the fracture height is influenced by the reservoir thickness, the fracture height increased only 5.3% with the increase of flow. The change in the fracture width is influenced by the stress difference inside and outside fractures, and the change in the fracture length is influenced by the loss conditions of fractures. When the fracture geometric extends to a threshold, it is necessary to adjust the injection parameters to further develop fractures.

Effect of sand ratio on fracture geometry

From the perspective of production parameters, a change in sand ratio considerably impacts productivity. Simultaneously, during the transportation of fluid in the reservoir, the pressure within fractures increases as the sand ratio increases. When setting simulation parameters, the addition of sand follows production guidelines. The sand ratio of the total amount injected is used as the marking point, it is also represented by the change rule of the main fracture parameters in the fracture network. In shale reservoirs with good natural fracture development and large fracture widths, the sand ratio can be reduced, but not <1%. For shale reservoirs with poor natural fracture development, the sand ratio can be increased, but not >10%, and the stage sand ratio should be kept <20%.

Figure 10 shows the change in the fracture structure at different sand ratios, and the changing mechanism was analyzed in conjunction with knowledge related to proppant transport. Simulation results show that a change in the sand ratio has a relatively small impact on the structure of fractures, with a length increase of 0.5%, a height increase of 0.04%, and a width increase of 1.1%. A comparison of the effects of the flow rate and sand ratio on the change in fracture geometry shows that fluid characteristics are the primary factors influencing the geometric structure of fractures in fracture.

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Effect of injection conditions on the conductivity of fracture networks

The analysis of the calculation process in the mathematical model shows that the conductivity of the fracture network is closely related to the structure and permeability of reservoir. Assuming a consistent distribution of the proppant within the fracture network, one can assume that the average permeability of secondary fractures in the fracture network is similar to that of the main fractures. Combining this assumption with the distribution of closure widths in the fractures, the conductivity of secondary fractures can be calculated.

Effect of injection method on fracture conductivity

The permeability of DFNs is influenced by reservoir and natural fracturing conditions, resulting in complex characteristics. Because the distribution and arrangement of fractures vary in different directions, permeability also varies in different directions, leading to permeability anisotropy. Although the proppant may create diversion within the fracture network, the main fractures remain the most crucial flow channels. Therefore, by simulating the distribution of the proppant in the main fractures, the impact of injection methods on the conductivity of fractures is analyzed.

The distribution of the proppant concentration in the main fractures under closure conditions is shown in Figure 11, the injection method considerably impacts the proppant distribution in fracture. Due to the long length of the fractures, the proppants in 3HF4 and 3-6HF1 cannot easily be transported to the toe of fractures. In 3HF4, the proppants form multiple aggregation areas in the fractures, but their settling is more pronounced. In 3-3HF1, the proppants are distributed more uniformly in the fractures, and due to the underdeveloped fracture network, the proppant concentration in the main fractures of 3-3HF1 is higher than those of 3HF4 and 3-6HF1. In 3-6HF1, the proppants form a distinct plug at the front end of the fracture, and the distribution of proppants is uniform at the rear end. Different injection methods result in different proppant distribution patterns, leading to differences in conductivity.

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Compared with Figure 11A–C, simulation results show that the intermittent injection of the proppant in fracture can control the distribution of sand plugs. During continuous injection, the proppant gradually accumulates within the fractures, forming sand plugs. By periodically stopping and restarting the injection of the fracturing fluid, the accumulation of sand plugs can be disrupted, allowing the proppant to distribute more uniformly within fractures. Moreover, during the process of injecting the stage-fracturing fluid, intermittent injection resuspends settled particles in the liquid, enhancing the uniform dispersion of proppant particles in fracture. By adjusting the frequency and duration of injecting the sand-carrying fluid, the distribution pattern of the proppant can be controlled. By optimizing proppant distribution patterns, intermittent injection can improve the efficiency of hydraulic fracturing.

From the results in Figure 12A, the improvement in the fracture conductivity is attributed to the uneven distribution of the proppant in the fracture, leading to the formation of numerous low-resistance channels within the propped fractures. The distribution of the proppant in the fracture network reveals that the proppant particle size at the front end of the fractures is small in Figure 12B,C, resulting in a lower conductivity. Intermittent injection changes the flow field within fractures, leading to irregularities in the distribution of the proppant.

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Using a convection model to simulate the distribution of the proppant within the propped fractures, different patterns of conductivity changes at various locations in the fractures can be obtained. The analysis of the average conductivity at different positions in the fractures shows that although the distribution of the proppant after intermittent injection is uneven, it macroscopically exhibits an intermittent distribution pattern. Because of the influence of the proppant distribution, there is a significant variation in the conductivity at different positions in fractures.

During continuous injection, the conductivity of fractures near the wellbore gradually decreases from near to far from the wellbore. During the intermittent injection process, the proppants are distributed in stages, causing the conductivity of fractures to fluctuate from near to far in a wavelike manner.

Effect of injection parameters on the conductivity of fracture networks

From the simulation results, the main fracture length and distance between fractures in the fracture network in the directions alone X- and Y-axes can be obtained. The parameters of secondary fractures can be calculated based on the calculation model described in Section 2.1. The process of calculating the diversion capacity of the fracture network is as follows: ① The average diversion capacity of the main fracture is determined based on the simulation results. ② Assuming that the seepage resistance of the unit fracture in the fracture network is the same, the seepage resistance of the secondary fracture is calculated using the conductivity of the main fracture and the fracture morphology parameters. ③ The diversion capacity of the fracture network was calculated using the calculation model in Section 2.2.

To validate the computational model, the experimental results from Wen et al.³² were used to compare the diversion capacities of different structured fracture networks. The experimental process assumed a regular fracture with no influence from closure stresses, conditions applicable to the conductivity calculation model used in this study. It was assumed that the movement of the proppant in secondary fractures is consistent with that in the main fractures. The calculated results are shown in Figure 13.

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Figure 13 shows that the calculated results are slightly smaller than the experimental data, considering the conditions analyzed in the experiment and model data. On the one hand, this is because the experiment used gas-measured permeability, leading to slightly lower calculation values of the conductivity. On the other hand, the distribution of the proppant during the experiment was relatively complex, but overall, the changing trend is generally consistent. According to the literature, the complexity of the fracture network is positively correlated with the conductivity. On the basis of simulation results under different injection conditions, the conductivity of the fracture network was calculated.

Figure 14 shows that as the flow rate increases from 12 to 22 m3/min, the average conductivity exponentially for 3HF4 decreases by 50.87%, 3-3HF1 decreases by 36.72%, and 3-6HF1 decreases by 41.06%. The primary reason for this phenomenon is that an increase in the flow rate leads to an enlargement of the fracture volume with constant total injection, consequently reducing the proppant concentration per unit volume in fracture. During continuous injection, the sand bank primarily extends through translation, whereas during intermittent injection, the proppant is suspended and dispersed. Therefore, an increase in the flow rate has a significant impact on intermittent injection.

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The sand ratio is an important factor affecting the flow capacity of fractures, and the simulation neglects the effects of the particle size changes and embedding fractures. Figure 15 shows that the conductivity for 3HF4 decreases by 4.34%, 3-3HF1 decreases by 3.56%, and 3-6HF1 decreases by 4.13% with an increase in the sand ratio. In addition to the effect of fracture volume change, sand ratio changes have other effects. A comparison of the distribution of the proppant in fracture at different sand ratio conditions shows that a higher sand ratio may lead to the excessive accumulation of the proppant in fractures, resulting in the formation of continuous sand dams and reducing the effective pathways of fractures. Additionally, an excessive amount of the proppant increases the pressure in fractures, leading to fracture closure. In summary, the flow capacity decreases as the sand ratio increases because the sand flow restricts the pathways for the fluid flow through fractures and increases fluid resistance.

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Compared with the volume of the fracture geometry parameters, the flow capacity of the fracture network is more significantly influenced by the injection method. Increasing the density of the fracture network generally enhances the overall flow capacity because a dense network of fractures more effectively transmits the fluid. Wider fractures and higher porosity typically result in higher permeability, thereby increasing the flow capacity. A clearer understanding of the changes and distribution patterns in flow capacity was obtained by analyzing the change in the flow capacity under intermittent injection conditions.

CONCLUSION

The structure of DFNs is strongly correlated with the changes in conductivity, where the density and connectivity of the fracture network are the primary determining factors of conductivity. Fracture networks with more fractures, higher density, and better connectivity typically exhibit higher diversion capacities. In this paper, the maximum difference in fracture network volume between models is 10.87%, and the maximum difference in SRV is 69.35%.

The structure of the fracture network considerably impacts the conductivity, influenced by the development of natural fractures and geological conditions. The width of fractures affects the flow velocity of fluids, and the length of the fractures determines the transport distance of fluids, thus influencing the conductivity of fractures. When the injection volume is kept constant, when the injection flow rate increases by 83.3%, the average fracture length increases by 14.6%, the width decreases by 25.5%, the height increases by 5.3%, and the diversion capacity decreases by 42.9%.

The proppant distribution is an important factor influencing the fracture conductivity, and the intermittent injection of the proppant can cause significant changes in the distribution pattern. When the sand ratio is doubled, the average fracture length increases by 0.5%, the height increases by 0.04%, the width increases by 1.1%, and the conductivity decreases by 4.3%. Increasing the sand ratio in the intermittent injection process will form a continuous distribution of sand banks and reduce high-speed flow channels. Therefore, during the fracturing process, techniques such as physical mixing and fiber addition are combined to enhance the propping effect of dominant channels.

The structure of the fracture network is related to the reservoir property. In this paper, the calculation method of fracture network conductivity derived is suitable for shale reservoirs with relatively developed natural fractures. The calculation model of the diversion capacity involves the fracture network expansion and the principle of hydroelectric similarity, based on which the diverting criterion of the branch fracture can be further developed.

AUTHOR CONTRIBUTIONS

The manuscript was written through the contributions of all authors. All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENTS

This study was funded by the China National Natural Science Foundation (Grant No. 52174056).

CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.

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